Models for Mapping Class Groups II

A celebrated theorem of Nikolai Ivanov states that the automorphism group of the mapping class group is again the mapping class group. The key ingredient is his theorem that the automorphism group of the complex of curves is the mapping class group. After many similar results were proved, Ivanov made a metaconjecture that any “sufficiently rich object” associated to a surface should have automorphism group the mapping class group. In joint work with Tara Brendle, we show that the typical normal subgroup of the mapping class group (with commuting elements) has automorphism group the mapping class group. To do this, we show that a very large family of complexes associated to a surface has automorphism group the mapping class group.